A Limit Theorem for a Class of Interacting Particle Systems

Abstract

Let $S$ be a countable set and $\Lambda$ the collection of all subsets of $S$. We consider interacting particle systems (IPS) $\{\eta_k\}$ on $\Lambda$, with duals $\{\tilde\eta_t\}$ and duality equation $P\lbrack |\eta^\zeta_t \cap A| \operatorname{odd} = \tilde{P}\lbrack |\tilde\eta_t^A \cap \zeta| \operatorname{odd} \rbrack, \zeta, A \subset S, A$ finite Under certain conditions we find all the extreme invariant distributions that arise as limits of translation invariant initial configurations. Specific systems will be considered. A new property of the annihilating particle model is then used to prove a limiting relation between the annihilating and coalescing particle models.